Algunas aportaciones de la investigación operativa a los problemas de localización

La toma de decisiones sobre localizaciones atrae, por su impacto social y económico, creciente interés de geógrafos, economistas y matemáticos. En las páginas que siguen describimos algunas aportaciones que se están realizando desde las Matemáticas (más concretamente, desde la Investigación Operativa), tanto en el modelado, como en la resolución de los problemas de Análisis de Localizaciones

Problemas de clasificación y optimización

El desarrollo de técncias que permitan clasificar entes (seres vivos, elementos, problemas, ... ) en distintas categorías ha sido recurrente en diversas ramas del saber. La creación y difusión de grandes bases de datos y la consiguiente necesidad de extraer conocimiento de las mismas han revitalizado el interés de la comunidad científica por tales técnicas. En estas páginas se ilustra cómo la Programación Matemática puede contribuir al diseño de métodos autoináticos de clasificación y profundizar en el conociiniento teórico de los mismos. Nuestra intención no es hacer una revisión cotnpleta del estado del arte en el tema, sino más bien describir someramente las aportaciones que en esle campo se están realizando en el seno del grupo PAi FQM-809 y en el proyecto de investigación BFM2002-04525-C02-02 del MCYT.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Ciencia y Tecnologí

Location and design of a competitive facility for profit maximisation

A single facility has to be located in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, and consumers patronise the facility to which they are attracted most. Attraction is expressed by some function of the quality of the facility and its distance to demand. For existing facilities quality is fixed, while quality of the new facility may be freely chosen at known costs. The total demand captured by the new facility generates income. The question is to find that location and quality for the new facility which maximises the resulting profits. It is shown that this problem is well posed as soon as consumers are novelty oriented, i.e. attraction ties are resolved in favor of the new facility. Solution of the problem then may be reduced to a bicriterion maxcovering-minquantile problem for which solution methods are known. In the planar case with Euclidean distances and a variety of attraction functions this leads to a finite algorithm polynomial in the number of consumers, whereas, for more general instances, the search of a maximal profit solution is reduced to solving a series of small-scale nonlinear optimisation problems. Alternative tie-resolution rules are finally shown to result in ill-posed problems.Dirección General de Enseñanza Superio

Optimal expected-distance separating halfspace

One recently proposed criterion to separate two datasets in discriminant analysis, is to use a hyperplane which minimises the sum of distances to it from all the misclassified data points. Here all distances are supposed to be measured by way of some fixed norm, while misclassification means lying on the wrong side of the hyperplane, or rather in the wrong halfspace. In this paper we study the problem of determining such an optimal halfspace when points are distributed according to an arbitrary random vector X in Rd,. In the unconstrained case in dimension d, we prove that any optimal separating halfspace always balances the misclassified points. Moreover, under polyhedrality assumptions on the support of X, there always exists an optimal separating halfspace passing through d affinely independent points. It follows that the problem is polynomially solvable in fixed dimension by an algorithm of O(n d+1) when the support of X consists of n points. All these results are strengthened in the one-dimensional case, yielding an algorithm with complexity linear in the cardinality of the support of X. If a different norm is used for each data set in order to measure distances to the hyperplane, or if all distances are measured by a fixed gauge, the balancing property still holds, and we show that, under polyhedrality assumptions on the support of X, there always exists an optimal separating halfspace passing through d − 1 affinely independent data points. These results extend in a natural way when we allow constraints modeling that certain points are forced to be correctly classified.Ministerio de Ciencia y Tecnologí

Continuous location problems and Big Triangle Small Triangle: constructing better bounds

The Big Triangle Small Triangle method has shown to be a powerful global optimization procedure to address continuous location problems. In the paper published in J. Global Optim. (37:305–319, 2007), Drezner proposes a rather general and effective approach for constructing the bounds needed. Such bounds are obtained by using the fact that the objective functions in continuous location models can usually be expressed as a difference of convex functions. In this note we show that, exploiting further the rich structure of such objective functions, alternative bounds can be derived, yielding a significant improvement in computing times, as reported in our numerical experience.Ministerio de Educación y CienciaJunta de Andalucí

Solving the median problem with continuous demand on a network

Where to locate one or several facilities on a network so as to minimize the expected users-closest facility transportation cost is a problem well studied in the OR literature under the name of median problem. In the median problem users are usually identified with nodes of the network. In many situations, however, such assumption is unrealistic, since users should be better considered to be distributed also along the edges of the transportation network. In this paper we address the median problem with demand distributed along edges and nodes. This leads to a globaloptimization problem, which can be solved to optimality by means of a branch-and-bound with DC bounds. Our computational experience shows that the problem is solved in short time even for large instances.Ministerio de Educación, Cultura y DeporteJunta de AndalucíaEuropean Regional Development Fun

Heliostat field cleaning scheduling for Solar Power Tower plants: a heuristic approach

Soiling of heliostat surfaces due to local climate has a direct impact on their optical efficiency and therefore a direct impact on the productivity of the Solar Power Tower plant. Cleaning techniques applied are dependent on plant construction and current schedules are normally developed considering heliostat layout patterns, providing sub-optimal results. In this paper, a method to optimise cleaning schedules is developed, with the objective of maximising energy generated by the plant. First, an algorithm finds a cleaning schedule by solving an integer program, which is then used as a starting solution in an exchange heuristic. Since the optimisation problems are of large size, a p-median type heuristic is performed to reduce the problem dimensionality by clustering heliostats into groups to be cleaned in the same period.Ministerio de Economía y Competitivida

Semi-obnoxious location models: a global optimization approach

In the last decades there has been an increasing interest in environmental topics. This interest has been reflected in modeling the location of obnoxious facilities, as shown by the important number of papers published in this field. However, a very common drawback of the existing literature is that, as soon as environmental aspects are taken into account, economical considerations (e.g. transportation costs) are ignored, leading to models with dubious practical interest. In this paper we take into account both the environmental impact and the transportation costs caused by the location of an obnoxious facility, and propose as solution method of the well-known BSSS, with a new bounding scheme which exploits the structure of the problem.Dirección General de Investigación Científica y Técnic

Supervised classification and mathematical optimization

Data Mining techniques often ask for the resolution of optimization problems. Supervised Classification, and, in particular, Support Vector Machines, can be seen as a paradigmatic instance. In this paper, some links between Mathematical Optimization methods and Supervised Classification are emphasized. It is shown that many different areas of Mathematical Optimization play a central role in off-the-shelf Supervised Classification methods. Moreover, Mathematical Optimization turns out to be extremely useful to address important issues in Classification, such as identifying relevant variables, improving the interpretability of classifiers or dealing with vagueness/noise in the data.Ministerio de Ciencia e InnovaciónJunta de Andalucí

Dominating sets for convex functions with some applications

A number of optimization methods require as a rst step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity. In this note we address the problem of constructing dominating sets for problems whose ob jective is a componentwise nondecreasing function of (possibly an in nite number of ) convex functions, and we show how to obtain a convex dominating set in terms of dominating sets of simpler problems. The applicability of the results obtained is illustrated with the statement of new localization results in the elds of Linear Regression and Location.Dirección General de Investigación Científica y Técnic